GIFT  OF 


^^*"*^ 


OSCILLATING -CURRENT 
CIRCUITS 

AN  EXTENSION  OF  THE  THEORY  OF  GENERALIZED 

ANGULAR   VELOCITIES,   WITH   APPLICATIONS   TO 

THE    COUPLED    CIRCUIT    AND    THE    ARTIFICIAL 

TRANSMISSION  LINE 


BY 

V.  BUSH 


ABSTRACT 

OF 
A  THESIS 

SUBMITTED  TO  THE  FACULTY  OF  THE 
MASSACHUSETTS  INSTITUTE  OF  TECH- 
NOLOGY IN  PART  FULFILMENT  OF 
THE  REQUIREMENTS  FOR  THE  DEGREE 
OF  DOCTOR  OF  ENGINEERING 


JUNE,  1916 


OSCILLATING-CURRENT 
CIRCUITS 

AN  EXTENSION  OF  THE  THEORY  OF  GENERALIZED 

ANGULAR  VELOCITIES,   WITH    APPLICATIONS   TO 

THE    COUPLED    CIRCUIT    AND    THE    ARTIFICIAL 

TRANSMISSION  LINE 

BY 

V.  BUSH 


ABSTRACT 

OF 

A  THESIS 

SUBMITTED  TO  THE  FACULTY  OF  THE 
MASSACHUSETTS  INSTITUTE  OF  TECH- 
NOLOGY IN  PART  FULFILMENT  OF 
THE  REQUIREMENTS  FOR  THE  DEGREE 
OF  DOCTOR  OF  ENGINEERING 


JUNE,   1916 


CONTENTS. 

PAGE 

LIST  OF  SYMBOLS  EMPLOYED »    .    .  4 

INTRODUCTION 5 

THE  COUPLED  CIRCUIT 8 

APPLICATION  TO  THE  ARTIFICIAL  LINE 1 1 

SUGGESTIONS  FOR  A  CONTINUATION  OF  THE  WORK 14 

SUMMARY 14 


333554 


LIST  OF  SYMBOLS  EMPLOYED  IN  THESIS. 

i  The  instantaneous  oscillating  current  in  a  branch  of  a  network — am- 
peres 

n  A  generalized  angular  velocity  of  oscillation — hyperbolic  radians  per 
second  Z 

Z    Generalized  impedance — ohms  Z 

E    Initial  potential — volts 

€     Naperian  base — 2.718   .... 

twi«2  Roots  of  the  equation  Z  =o,  hyperbolic  radians  per  second  Z 

C    Total  capacitance — farads 

L    Total  self  inductance — henries 

M  Total  mutual  inductance — henries 

R    Total  resistance — ohms 

Constants  of  the  primary  and  secondary  of  a  coupled  circuit  are  dis- 
„        tinguished  by  subscripts 

a,  /3,  7,  5,  77     Coefficients  of  the  equation  Z  =  o  for  the  coupled  circuit 

q  Correction  to  be  applied  to  the  absolute  values  of  the  free  angular 
velocities  of  a  resistanceless  coupled  circuit  to  obtain  the  absolute 
values  of  the  angular  velocities  of  the  complete  circuit.  numeric 

p    A  correction  to  be  added  and  subtracted  to  -  to  obtain  the  decrements 

4 
of  the  complete  coupled  circuit. •  hyp.  rad.  per  sec. 

s,  t  Sum  and  difference  respectively  of  the  squares  of  the  angular  velocities 
of  the  resistanceless  coupled  circuit. 


/hyp.    rarl    \2 


>.  rad.V 
sec.      / 


\      sec 

j     The  pure  imaginary,  V  —  i 

A    A  generalized  amplitude  of  current  oscillation — amperes  Z 
m    Number  of  sections  of  an  artificial  line 
h     Auxiliary  constant  numeric 

.Z  This  sign  appended  to  the  units  of  an  equation  denotes  that  the  expres- 
sion contains,  in  general,  complex  quantities 


OSCILLATING-CURRENT  CIRCUITS. 


INTRODUCTION. 

Heaviside,*  and  since  then  several  others,!  have  shown  that  for  the  free 
oscillations  of  a  network  the  generalized  impedance,  formed  from  the  con- 
stants of  the  network  and  the  complex  angular  velocity  of  oscillation,  is 
zero  for  any  complete  circuit.  This  principle  enables  the  frequencies  and 
decrements  of  the  free  oscillations  of  a  network  to  be  readily  found.  There 
is  a  similar  principle  which  enables  the  finding  of  the  amplitudes  of  free 
oscillation  at  the  several  frequencies,  which  is  also  in  Heaviside,  derived 
from  a  series  of  theorems  concerning  the  distribution  of  energy  during 
subsidence.  It  is  the  purpose  of  the  thesis,  of  which  this  is  an  abstract,  to 
demonstrate  the  application  of  this  second  principle  to  practical  engineering 
problems. 

The  principle  may  be  stated  as  follows:  If  Z  is  the  generalized  impedance 
of  a  branch  of  the  network  initially  containing  a  store  of  energy,  corre- 
sponding to  the  initial  voltage  E,  and  if  n  is  the  complex  angular  velocity 
of  oscillation,  so  that  Z=  /(«),  then  the  first  order  term  in  the  Taylor  ex- 

j     rj 

pansion  of  Z,  namely,  n  ,  will  be  of  the  nature  of  an  impulsive  impedance ; 

d  n 

and  the  oscillatory  current  will  be  of  the  form : 

E     nt  , 

€  amperes  Z 


--^   dz 

n  — 
dn 

where  the  summation  extends  over  the  roots  «i,  n^,  -  -  of  the  equation  Z  =o. 

It  will  be  convenient  to  call  the  expression  n the  "threshold  im- 

d  n 
pedance."* 

The  equation,  as  given,  applies  to  the  current  in  the  branch  initially 
charged,  where  the  generalized  and  threshold  impedances  are  formed  for 
that  branch. 

The  discussion  of  the  application  of  this  principle  to  various  typical  net- 
works has  indicated  the  truth  of  the  following  additional  propositions 
which  will  be  found  useful  in  attacking  particular  problems: 

*  Heaviside.     Electrical  Papers,  Electromagnetic  Theory,  Vol.  II. 
t  Campbell,  Proc.  AIEE,  1911; 

Kennelly,  Proc.  IRE,  1915; 

Eccles  &  Makower,  Electrician,  1915. 


(1)  In  determining  the  amplitude  of  oscillation  at  some  point  of  the  net- 
work distant  from  the  branch  initially  charged,  the  generalized  impedances 
of  the  elements  combine  in  the  manner  of  simple  resistances.     Upon  com- 
bining with  the  generalized  impedance  of  an  element,  each  term  of  a  cur- 
rent or  voltage  expression  is  combined  with  the  generalized  impedance  of 
the  element  formed  for  the  free  angular  velocity  of  the  term  considered. 

(2)  When  several  stores  of  energy  are  simultaneously  discharged  they 
may  be  considered  separately  and  the  results  added. 

(3)  In  order  to  ensure  that  the  correct  free  angular  velocities  be  ob- 
tained, the  generalized  impedance  should  be  formed  for  the  branch  under 
examination;  as  in  special  cases  certain  free  angular  velocities  may  be 
absent  in  particular  branches  of  the  network. 

(4)  The  threshold  impedance  is  formed  always  from  the  generalized 
impedance  which  considers  the  initially  charged  element  as  the   main 
branch. 

(5)  The  sudden  application  of  a  steady  electromotive  force  may  be 
treated  as  the  inverse  of  the  discharge  from  the  final  state  attained. 

(6)  The  sudden  application  of  an  alternating  electromotive  force  may  be 
treated  in  similar  manner,  the  unbalanced  stores  of  energy  being  in  this 
case  the  differences  between  the  initial  stores  of  energy  in  the  network,  and 
the  energies  at  the  same  points  of  the  network  corresponding  in  the  steady 
state  to  the  point  of  the  voltage  wave  at  which  it  was  suddenly  applied. 

The  method  of  applying  the  threshold  impedance  is  shown  by  various 
examples.  One  of  these,  the  series  circuit  containing  resistance,  inductance, 
and  capacitance  is  included  here  for  illustration. 


Resistance 
f\   o/ims. 


Inductance 
L_     henries. 


C     farads. 


P/g.  / .       5imf)le      Series     Osci/latin<j     Circuit. 


• 
7 

In  this  circuit  (see  fig.  i)  the  generalized  impedance  will  be: 

Z  =  R+Ln+—  ohms  Z 

Cn 

Equaling  to  zero  and  solving  for  n,  we  obtain  the  free  angular  velocities: 


L 

LC 


_  R     .  //  R\*     r 

71%  —  —  —          %  /    I    -   I     —  - 

*L       V  \2L/       LC 


hyp-  rad. 

sec. 


The  threshold  impedance  is: 


n —  =Ln—  —  ohms  Z 

dn  Cn 

If  now  we  consider  the  condenser  as  discharging  through  the  circuit  from 
an  initial  voltage  E,  the  current  will  be: 

n=nz     E      ni  , 

i=   «^  e  amperes  Z 

n^x   n^ 
H  dn 

or 

E          nit  i         E          n%t  / 

t  = e      -j- e  amperes  Z. 

i  _     i 


which,  with  the  values  of  n\  and  n*  given  above,  is  the  complete  oscillatory 
solution.  This  expression  may  be  reduced  to  the  usual  form  by  inserting 
the  values  of  «i  and  n^.  There  will,  of  course,  be  three  cases  according  as 
the  quantity  under  the  radical  is  positive,  zero,  or  negative.  For  the  third 
case  the  expression  becomes  upon  reducing: 


amperes 
LC     \2L' 


which  is  the  solution  obtained  by  the  usual  methods.     The  solutions  for 
the  other  cases  may  be  obtained  by  similar  reductions. 

It  will  be  noted  that  this  method  of  solving  the  circuit  is  much  more  con- 
cise and  direct  than  is  the  method  of  determining  the  constants  of  integra- 
tion in  the  differential  equation  solution,  in  accordance  with  the  boundary 
conditions.  It  is  also  convenient  to  retain  all  three  cases  in  the  single 
expression. 


8 

If  we  wish  the  oscillatory  voltage  across,  for  instance,  the  reactor  in  this 
circuit,  we  may  obtain  it  by  multiplying  the  oscillatory  current  by  the 
generalized  impedance  of  the  reactor,  and  treat  the  current  terms  separately, 
thus: 

nit  ,       ELnz        n2t  , 

-  e      + €  volts  Z 


i  i 

Cn\  Cn2 

and  this  expression  may  also  be  reduced  by  inserting  the  values  of  n\  and  n^. 

THE  COUPLED  CIRCUIT. 

The  coupled  circuit  has  been  thoroughly  solved  by  the  method  of  differ- 
ential equations.*  These  solutions  have  been  discussed  from  the  point  of 
view  of  the  applications  of  this  circuit,  particularly  to  radio  work.  Many 
approximate  solutions  have  been  obtained  for  the  case  of  the  free  discharge 
of  the  primary  condenser,  either  by  neglecting  the  effects  of  resistance,  or 
the  reaction  of  the  secondary  upon  the  primary,  or  in  some  similar  way. 
The  complete  exact  solution  has  been  generally  avoided,  principally  because 
of  its  complication.  The  resistance  operator  method,  or  the  method  of 
generalized  angular  velocities,  has  also  been  applied  to  this  circuit  in  as  far 
as  the  frequencies  and  decrements  are  concerned.!  This  method  gives  the 
same  equation  for  the  determination  of  the  free  angular  velocities  as  does 
the  differential  equation  solution,  namely: 


numerc 


where  n  is  the  complex  angular  velocity,  and  the  constants  are  those  shown 
on  fig.  2. 


Fi<j.i.     JndvctiVely     Counted     Circuit. 

*  Bjerkness,  Wied.  Ann.  55,  1895;  Oberbeck,  Wied.  Ann.  55,  1895;  Domalys  &  Kolacek, 
Wied.  Ann.  57,  1896;  Wien,  Wied.  Ann.  61,  1897;  Rayleigh,  Theory  of  Sound;  Braun,  Phys. 
Za.  3,  1901;  Drude,  Ann.  d.  Phys.  13,  1904;  Jones,  Phil.  Mag.  1907;  Cohen,  Bui.  Bu.  Stds. 
S,  1909;  Pierce,  Proc.  Am.  Ac.  A.  &  Sc.  46, 1911;  Fleming,  Proc.  Phys.  Soc.  1913- 

t  Eccles,  Phys.  Soc.  Proc.  24,  1912.     Kennelly,  Proc.  IRE,  1915. 


The  solution  of  this  fourth  degree  equation  is  laborious,  and  may  be 
avoided  in  the  following  manner.    The  equation  may  be  written  in  the  form : 

/hyp.  rad.X4   , 


and  if  we  treat  the  same  circuit  without  resistance  we  obtain  the  easily 
solved  equation: 


/hyp-  rad.V 

I  -^-  -  ) 

\      sec.      / 


The  roots  of  this  last  equation  will  differ  but  little  in  absolute  value  from 
the  absolute  values  of  the  roots  of  the  complete  equation.  If  (i  -\-q)  and 
(i  —  q)  are  the  correction  factors  to  be  applied  to  the  absolute  values  of  the 
resistanceless  roots  in  order  to  obtain  the  absolute  values  of  the  complete 
roots,  we  may  find  an  expression  for  q  by  means  of  the  algebraic  relations 
between  the  roots  and  coefficients  of  the  above  equations;  and  in  deriving 
this  relation  the  square  of  q  may  be  neglected.  This  expression  is: 

P-yt-ctS 

-  -  -  numeric 


where  s  is  the  sum  and  /  the  difference  of  the  squares  of  the  roots  of  the 
resistanceless  equation. 

In  a  similar  manner  the  relation  may  be  derived: 

27— as+aqt 
4t—8qs 

where  (  -+p  }  and  (  -— p  }  are  the  decrements  in  the  solution  of  the  com- 
\4       /  \4       / 

plete  circuit.  In  this  manner  the  frequencies  and  decrements  of  the  oscil- 
lations in  the  coupled  circuit  may  be  obtained  without  the  necessity  of 
solving  the  fourth  degree  equation. 

This  method,  tested  on  a  typical  circuit  with  constants: 

Ci  =  io~9  farads 
C2  =  io-10  farads 
RI  =1000  ohms 
RZ  =2000  ohms 
LI  =0.025  henries 
L2  =0.040  henries 
M  =0.020  henries 

gave  by  exact  solution: 

—  17638.9=117192683. 


10 

and  by  the  approximate  method  : 

-  57361.  o±;  664750. 
-17639.0^192680. 

The  amplitudes  of  oscillation  may  be  readily  found  by  the  use  of  the 
threshold  impedance.  If  we  consider  the  discharge  of  the  primary  con- 
denser, so  that  the  primary  is  the  main  branch,  the  threshold  impedance  is: 


n 


ohms  Z 


Inserting  into  this  expression  the  four  roots  MI,  w2,  w3,  w4  of  the  equation  Z  =  o, 
gives  four  particular  values  of  the  threshold  impedance.  Dividing  the 
initial  primary  condenser  voltage  by  each  of  these  values  gives  the  four 
complex  amplitudes  for  the  primary  current  expression: 

t    nit  ,        n%t  .    .    nzt  .    .    w4/  / 

^l  =  Al€      +A2e      +A3e      +  Ate     ,  amperes  L 

This  expression  may  be  readily  reduced  to  trigonometric  form,  when  the 
imaginary  portions  of  the  expression  cancel  out. 

An  examination  of  the  generalized  impedance  of  the  several  elements  of 
the  circuit  gives  for  the  ratio  between  the  primary  and  secondary  ampli- 
tudes: v 

Mn  .     / 

—  -  numeric  Z. 


The  four  values  of  this  ratio  applied  to  the  primary  amplitudes  give  the 
corresponding  secondary  amplitudes* 

The  results  above  were  checked  by  means  of  oscillograms  taken  upon 
a  typical  coupled  .circuit.     The  constants  chosen 

^i  =I-937  ohms 

Rz  =2.531  ohms 

LI  =7.52  x  io~3  henries 

Lt  =7.63  x  i  o~3  henries 

M  =3475  x  io~3  henries 

C\  =13.51  microfarads 

€2  =24.62  microfarads 

gave  frequencies  of  oscillation  609.5  °°  and  339-2  °°  which  were  within  con- 
venient range  for  the  oscillograph.  The  computed  points  checked  the  os- 
cillograms within  the  errors  of  measurement.  A  solution  was  also  made  by 
differential  equations  as  a  check. 


II 

APPLICATION  TO  THE  ARTIFICIAL  LINE. 

There  has  been  much  difficulty  encountered  in  the  analysis  of  smooth  line 
transients.  The -cable  has  been  comparatively  easily  handled,*  but  the 
analysis  of  the  aerial  line  has  given  in  general  results  too  complicated  for 
engineering  use.  For  experimental  analysis  for  steady  state  phenomena 
the  lumped  artificial  line  has  proved  invaluable,!  but  there  has  been  much 
doubt  as  to  just  how  far  such  a  line  of  a  given  number  of  sections  could  be 
trusted  for  transient  effects,  t 

The  method  of  generalized  angular  velocities  is  applied  in  the  thesis  to 
the  analysis  of  the  oscillations  of  the  artificial  line  under  certain  typical  con- 
ditions. The  distant-end  current  on  a  grounded  artificial  line,  when  a 
steady  voltage  is  suddenly  applied  at  the  home  end,  is  considered  for  the 
artificial  cable,  and  the  artificial  aerial  line.  The  IT  line  is  used,  but  the 
formulas  apply  also  to  the  T  line  with  small  changes. 

The  purpose  of  this  analysis  is  to  determine,  for  specific  cases,  the  number 
of  sections  requisite  in  an  artificial  line,  in  order  that  it  may  represent  its 
corresponding  smooth  line,  not  only  for  the  steady  state,  but  also  for  certain 
transient  effects,  to  a  sufficient  degree  of  approximation  for  engineering 
investigations. 

The  method  used  is  simply  to  analyze  artificial  lines  with  various  num- 
bers of  sections,  considered  simply  as  networks  with  concentrated  constants. 
The  results  of  these  successive  solutions  are  grouped,  and  from  them  is 
derived  the  solution  for  the  general  case  of  m  sections. 

The  general  solutions  for  the  application  of  a  steady  electromotive  force 
to  the  grounded  artificial  line  obtained  in  this  way  follow: 

For  the  cable  of  m  sections  containing  resistance  and  capacitance  only: 

[—  m*hit  m*fe*  "~| 

<«„*     I+!^=l(  RC  -^Z2e  RC  +   .   .  .  amperes 

RL  22  J 

where 

*#  is  the  received  current 
E  the  applied  steady  voltage 
R  the  total  resistance 
Cthe  total  capacitance. 

*  Kelvin,  Proc.  Roy.  Soc.  1855; 

Poincar6,  EC.  Elect.  40,  1904; 

Malcolm,  Electrician  1911;  12. 
fPupin,  Trans.  AIEE  1890,  1900;  Trans.  Am.  Math.  Soc.  1900; 

Kennelly,  Proc.  Am.  Acad.  Arts  &  Sci.,  44,  1908 ; 
.  Huxley.  Thesis  M.  I.  T.,  1914. 
J  Cunningham  and  Davis,  Proc.  AIEE  1911,  1912; 

Ricker.  Thesis  M.  I.  T.  1915. 


12 

The  values  of  h  are  found  as  roots  of  the  auxiliary  equation: 


=0 


2  ! 

numeric 


where  there  are  —  terms  if  m  is  even,  and terms  if  m  is  odd. 


A  curve  for  obtaining  these  roots  is  presented  for  convenient  use  in  prac- 
tical cases.  , 

For  the  aerial  line  containing  resistance,  capacitance  and  inductance, 
the  corresponding  equation  is: 


r 

<-![- 


LC 


hm—2 —  ^ 

2L  2L 


— I 

J      m'fe      (  R\* 

V  ~7C  "  (71  ^  +    '   '   J 


LC       \*  ^/  amperes 

As  the  line  is  subdivided  an  oscillatory  term  appears  in  this  equation  for 
each  section  introduced. 

This  aerial  line  formula  was  checked  by  means  of  oscillograms  taken 
upon  a  typical  artificial  aerial  power  transmission  line  at  Pierce  Hall, 
Harvard  University.*  Twelve  sections  were  used,  representing  a  line  of 
the  following  constants: 

#000  A.W.G.  aluminum  stranded  conductors 

Overstrand  diameter  0.47  inches 

Interaxial  distance  90.5  inches 

Length  596.4  miles. 

Total  inductance  1.035  henries 

Total  capacitance  8.ioXio~6  farads 

Total  resistance  300. 1  ohms 

The  elements  of  this  artificial  line  were  grouped  in  such  a  manner  that  it 
was  arranged  as  a  IT  line  of  various  numbers  of  sections.  The  arrival  curve 
of  current  computed  and  plotted  to  the  scale  of  the  oscillograms  showed  a 
check  to  a  reasonable  degree  of  accuracy. 

*  Kennelly  and  Tabossi,  Elec.  World  1912. 


In  order  to  determine  the  relation  between  the  artificial  and  smooth 
cables,  the  cable  formula  was  plotted  for  the  numerical  case: 

£=200  ohms 

C=  io~6  farads 

R=2O  volts 

for  various  numbers  of  sections.  It  was  found  that  with  these  constants, 
the  arrival  curve  on  a  six  section  artificial  cable  coincided  to  a  sufficient 
degree  of  accuracy,  for  the  purposes  of  engineering,  with  the  smooth  line 
arrival  curve  as  plotted  from  Kelvin's  formula: 


-C- 


2  e 


•n-n 
'RC 


RC 


amperes 


The  limiting  value  of  the  artificial  aerial  line  formula  as  the  number  of 
sections  is  indefinitely  increased  was  also  considered.  From  the  fact  that 
the  artificial  cable  formula  approaches  Kelvin's  smooth  line  formula  in  the 
limit,  were  derived  the  limits  of  the  various  values  of  mzh^  and  h  as  m=  °o  . 
A  certain  approximation  was  also  made  because  of  the  fact  that  in  lines 


encountered  in  practice 


/  R 
ice,  I  - 

\2  Li 


L 
may  be  neglected  in  comparison  with  -  . 

X/C 

Applying  these  facts  to  the  artificial  aerial  line  formula  gave  the  following 
expression  for  the  received  current  on  a  grounded  smooth  aerial  line  when  a 
steady  voltage  is  suddenly  applied  at  the  home  end  : 


Rt 


Rt 


Er 

*-*-' 


- € 


amperes 


where  F  (/)  is  the  discontinuous  function  represented  in  fig.  3. 


HLC-+ 


fig.  3.  The 


Fft)   in  tfte  atriaL  tine  formula. 


14 

The  arrival  curve  plotted  from  this  formula  for  the  smooth  line  on  which 
the  oscillograms  were  taken  is  shown  in  fig.  4. 


-4-£"ft/W 


Time 


Fig. 4.     Arrival    Current,   Smooth  Aerlat  Line. 

A  comparison  of  this  smooth  line  curve  with  the  artificial  line  arrival 
curves  showed  that  the  artificial  line  of  four  sections,  or  less,  did  not  well 
approximate  the  smooth  line,  for  the  transient  due  to  the  sudden  application 
of  a  steady  voltage.  The  artificial  line  of  twelve  sections  approximated 
the  smooth  line  fairly  well;  but  a  still  greater  number  of  sections  would  be 
necessary,  in  order  to  enable  the  artificial  line  to  be  used  for  experimental 
investigation  with  this  type  of  transient. 

SUGGESTION    FOR    A    CONTINUATION    OF    THE    WORK. 

The  method  of  generalized  angular  velocities,  applied  to  the  oscillations 
of  networks  with  concentrated  constants,  has  proved  to  be  valuable  for 
engineering  purposes.  It  is  believed  that  the  same  method  may  be  profit- 
ably applied  to  networks  containing  branches  with  distributed  constants. 
A  starting  point  for  such  work  would  be  found  in  Heaviside's  application 
of  the  resistance  operator  to  the  smooth  line. 


SUMMARY. 

In  addition  to  the  theorem  which  determines  the  free  angular  velocities 
of  oscillation  of  a  network,  there  is  a  theorem  which  will  determine  the 
amplitudes.  This  theorem  involves  a  "threshold  impedance"  which  may 
be  formed  for  any  circuit  with  concentrated  constants,  and  which  enables 
the  amplitudes  of  oscillation  to  be  found  from  the  initial  potential  of  the 
unbalanced  energy. 


15 

An  application  of  this  method  to  the  coupled  circuit  gives  an  easily  ap- 
plied and  convenient  complete  solution  for  the  primary  condenser  dis- 
charge. 

Applied  to  the  artificial  line,  it  enables  the  lumped  line  of  a  given  number 
of  sections  to  be  compared  with  the  represented  smooth  line  for  certain 
transient  effects. 

The  writer  wishes  to  express  his  thanks  to  Prof.  D.  C.  Jackson,  Dr.  A.  E. 
Kennelly,  and  other  members  of  the  Department  of  Electrical  Engineering 
who  have  assisted  him  in  the  preparation  of  the  thesis.  » 


R-6-i 6-400. 


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